Real-space renormalization-group study of the two-dimensional Hubbard model

Abstract

The one-band two-dimensional Hubbard Hamiltonian is analyzed using a real-space renormalization-group block method. The renormalization-group method, previously applied to the one-dimensional case has been extended to two dimensions. It is also shown how to avoid the proliferation of new terms during the process if the shape of the blocks and the symmetries of the kept states are chosen conveniently. We characterize the ground state by its energy per site, the gap of charge excitations, the double occupancy, and the effective hopping. The system shows an insulating behavior in the case of half filling for all nonzero interaction parameter, U, if the hopping is isotropic (${\mathit{t}}_{\mathit{y}}$=${\mathit{t}}_{\mathit{x}}$); that is to say, the gap opens for all positive U, as in the one-dimensional case. If the hopping anisotropy, \ensuremath{\alpha}=${\mathit{t}}_{\mathit{y}}$/${\mathit{t}}_{\mathit{x}}$, is different from the unity and zero the ground state is a conductor up to a critical value (U/t${)}_{\mathit{c}1}$, which depends on \ensuremath{\alpha}. We have analyzed in the same way another filling, n=7/9. The system behaves as a conductor, paramagnetic up to a value (U/t${)}_{\mathit{c}2}$\ensuremath{\sim}2 and weakly ferromagnetic for higher values of U.